Example

Solve for $x$: $\quad |2x+7| = |x-1|$

Solution

Equations with absolute value expressions on both sides are a little more complicated than our previous examples, but the ideas are the same.

A simple way to solve this type of equation is to use the definition of absolute value. Replace the absolute value bars with $\pm( \quad )$, and solve all equations that result from the different combinations of $+$ and $-$. Then you must check your solutions.

For this equation, we get

There are four equations to solve

but, in this case, the bottom two are equivalent to the top two. So we'll solve only the first two.

The first equation is

which has solution $x=-8$.

The second equation is

or

which has solution $x=-2$.

Our possible solutions are $x=-8$ and $x=-2$. You must check both of these in the original equation, and you will find that they both work. The solution set is